Large P4-free graphs with bounded degree

Let ex * (D; H) denote the maximum number of edges in a connected graph with maximum degree D and no induced subgraph isomorphic to H. We prove that this is finite only when H is a disjoint union of paths,m in which case we provide crude upper and lower bounds. When H is the four-vertex path P₄, we prove that the complete bipartite graph K_D,D is the unique extremal graph. Furthermore, if G is a connected P₄-free graph with maximum degree D and clique number ω, then G has at most D² ? D(ω ? 2)/2 edges. © 1993 John Wiley & Sons, Inc.